Visualizing Hot‐Carrier Expansion and Cascaded Transport in WS2 by Ultrafast Transient Absorption Microscopy

Abstract The competition between different spatiotemporal carrier relaxation determines the carrier harvesting in optoelectronic semiconductors, which can be greatly optimized by utilizing the ultrafast spatial expansion of highly energetic carriers before their energy dissipation via carrier–phonon interactions. Here, the excited‐state dynamics in layered tungsten disulfide (WS2) are primarily imaged in the temporal, spatial, and spectral domains by transient absorption microscopy. Ultrafast hot carrier expansion is captured in the first 1.4 ps immediately after photoexcitation, with a mean diffusivity up to 980 cm2 s−1. This carrier diffusivity then rapidly weakens, reaching a conventional linear spread of 10.5 cm2 s−1 after 2 ps after the hot carriers cool down to the band edge and form bound excitons. The novel carrier diffusion can be well characterized by a cascaded transport model including 3D thermal transport and thermo‐optical conversion, in which the carrier temperature gradient and lattice thermal transport govern the initial hot carrier expansion and long‐term exciton diffusion rates, respectively. The ultrafast hot carrier expansion breaks the limit of slow exciton diffusion in 2D transition metal dichalcogenides, providing potential guidance for high‐performance applications and thermal management of optoelectronic technology.

than that in the vertical direction. This leads the carrier shape from an approximate circle (0.2 ps) to ellipse (0.7 ps), nevertheless, the carrier shapes of bare WS 2 keep circular at every delay.
Since the grating gap (400 nm) is less than half of the spot size (~ 1.3 um), the non-uniformity of the whole structure can be ignored for the focused spot (also see morphology imaging of Figure S13d). Moreover, the energy position of A-exciton on the etched substrate is almost equal to that on the flat substrate (see Figure S13c), thus the strain effect caused by the grating can also be ignored. Therefore, the anisotropy is entirely due to the dielectric regulation of the grating. In the horizontal, the carrier transport proceeds smoothly in the semi-suspended WS 2 while it periodically experiences Si 3 N 4 and air substrate in the vertical. It can be inferred that the substrate dielectric regulation hinders carrier transport.
Also, for a quantitative description, the carrier transport parallel and perpendicular to the grating are further obtained by 1D asynchronous scanning, as shown in Figure S14b-e. At the initial 0.2 ps, the carrier spatial distribution in both directions is the same, but with time, the profile width in the parallel direction increases sharply, with an expansion rate significantly faster than that in the grating's vertical. The changes of linewidth extracted by Gaussian fitting are shown in Figure S14c, e, which shows a great difference. In the expansion stage, the average speed in the horizontal is 1300 cm 2 s -1 , while it is only 800 cm 2 s -1 in the vertical.
Compared with this, the speed on the uniform substrate is 980 cm 2 s -1 . Obviously, the grating not only hinders the hot carrier transport in the vertical but also gives the promotion in the horizontal. In addition, this anisotropy is also reflected in the contraction of the transition region. With the comparison between FigureS14c and e, the contraction amplitude of the vertical is significantly greater than that of the horizontal. Interestingly, in Discussion Section of the text we attribute the transition's contraction mainly to the speed mismatch between the hot carrier and the exciton, which is appropriate here because the large contraction in the vertical corresponds to the restricted motion of the exciton (the rate of the linear diffusion is nearly zero).
All in all, it can be concluded that the initial expansion of the transient absorption signal observed in the experiment comes from the ultrafast transport of hot carriers in real space.
This nonlinear and subsequent cascade transport can be regulated by the substrate.
Note S3. Thermo-optical cascaded carrier transport model Note S3a: Three-dimensional heat conduction We simulate the pump-induced carrier and lattice temperature changes in the carrier-latticesubstrate system. As mentioned in the text, the electron and lattice temperatures are calculated by the heat conduction equations as follows: [2,3] , | , where , , and are the temperature of the carrier, lattice, and substrate, respectively.
These temperatures are calculated using the finite element method, with the coefficients =2×10 3 J m -3 K -1 for the carrier heat capacity. [4] =1×10 5 J m -3 K -1 for the lattice heat capacity. [4] = 6.9×10 5 J m -3 K -1 for the Si 3 N 4 substrate heat capacity. [5] represents the normal direction, with zero positions at the WS 2 surface and 10 nm at the WS 2 -substrate interface. =( ) is the thermal conductivity tensor of the WS 2 carrier (lattice), where and is the in-plane and through-plane thermal conductivity. For inplane thermal conductivity, , with a simulated =105 W m -1 K -1 . [6] =31.8 W m -1 K -1 . [7] While for through-plane thermal conductivity, since the WS 2 thickness is only 10 nm, the specific value of has little influence on the temperature calculation results. So here, . [8] =90 W m -1 K -1 is the Si 3 N 4 thermal conductivity. [5] =2×10 15 W m -3 K -1 is the carrier-phonon coupling rate. [9] ( ) is the interfacial heat exchange rate between the carrier (lattice) and substrate, both set to be 1 MW m -2 K -1 . [10,11] is the initial heat source depending on the pump pulse, which can be defined as [12] √ where ~0.1 is pump absorbance at a photon energy of 3.1 eV, is the pump fluence varying from 8 to 80 μJ cm -2 , with an FWHM duration of =200 fs and spot width of =0.9 µm.
The penetration depth can be expected to be 10 nm according to an absorption coefficient of 1×10 8 m -1 . [13,14] For two limiting cases of the cascaded carrier transport, the initial hot carrier expansion rate can be roughly estimated to be . Upon pump excitation, stays almost the same, only increased by ~20 K, while drastically increase to thousands of Kelvin within 1 ps. Thus the fast expansion rate is roughly proportional to , which is well consistent with previous reports in semiconductor silicon. [15] In contrast, the long-term exciton diffusion is governed by the lattice thermal transport , which is approximately constant at different time delays due to the small change of and after 2 ps. [4] Note S3b: Permittivity calculation The temperature change is further reflected onto through thermo-optical conversion described below. Specifically, the photo-induced complex permittivity can be attributed to the free carrier (hot carrier) and bound exciton species, which is described using a Drude-Smith and Lorentz model, respectively. [16,17] where =15 is the fitted high-frequency contribution to the real part of . The second term represents the A-exciton Lorentzian contributions, where =4.2 is the fitted oscillator strength at resonance energy of =1.98 eV, and probe =2.06 eV is the probe photon energy. = • is the A-exciton broadening extracted from the linear absorption spectrum, which is proportional to the lattice temperature , with room temperature =293 K and corresponding broadening =35 meV. [17,18] The third term describes the Drude-Smith contributions from free carriers (hot carriers), where =5•10 -6 • is the Drude weight proportional to the density ratio ( ) between the hot carrier ( hot ) and exciton ( ex ). [16] It can be roughly estimated as hot ex b c under the assumption that thermal equilibrium is approached between the two excited states, [19] with an exciton binding energy of b 50 meV [20] for a 10 nm layered WS 2 and Boltzmann constant . probe is the probe angular frequency. ( ) is the coefficient determining the degree of carrier localization, with a near-zero value indicating a tendency to the Drude free-carrier limit, here C is set as 0.85. [16] is the carrier-carrier scattering time [21] that depends on carrier temperature, as discussed in the main text.

Note S3c: Acquisition of differential reflection signal
Finally, the photo-induced complex permittivity is transferred to optical reflectivity through Fresnel equation in a thin-film (air-WS 2 -Si 3 N 4 -Ag) under normal incidence are the reflectivity at air-WS 2 , WS 2 -Si 3 N 4 , Si 3 N 4 -Ag interfaces, respectively, with the refractive index of air =1, Si 3 N 4 thin film = 2 and Ag substrate g =0.057+4i for 2.06 eV probe. [22] e √ probe and e probe represent the optical path increments of the probe beam passing through WS 2 and Si 3 N 4 , with thicknesses of =10 nm and =180 nm and probe wavelength of 603 nm.
As a result, the transient differential reflectivity at specific time delays (t) and in-plane positions (x, y) can be expressed as (7) where represents the reflectivity of the sample before excitation at room temperature.
In our measurement time range, carrier-carrier scattering leads to the carrier thermalization and a drastic rising of the carrier temperature, resulting in a super-expansion of the hot carrier in the first ~1 ps. After that, carrier-phonon coupling leads to the decrease of carrier temperature and the formation of bound exciton at ~2 ps. Finally, the long-term excited state diffusion is dominated by the slow exciton diffusion, which is limited by the lattice thermal transport. By convolving them with the probe pulse and performing Gaussian fitting at different delay times, we can directly calculate the transport dynamics of the excited states and thus reproduce the experimental results.
Note S4. Cascaded carrier transport probed at the PB and low-energy PA resonance.
We note that the carrier diffusion dynamics probed at exciton resonance (PB feature at 1.98 eV) and low-energy PA resonance (1.92 eV) are completely different from that at high-energy PA feature (2.06 eV), as shown in Figure S8. According to the discussion in the main text, the high-energy PA feature is attributed to the hot carriers, while the PB and low-energy PA features are primarily caused by the band-edge bound excitons. Below, we will give a detailed discussion on the deviation of diffusion dynamics of the three features.
We firstly look at the spatial evolution of the PB feature, which represents the diffusion of the bound exciton. As shown in Figure S8a considered as an n-type semiconductor due to sulfur vacancy, which tends to capture holes).
Based on these two effects, the density ratio of hot carriers to excitons increases with the diffusion length, leading to a larger contraction of the exciton diffusion at the transition region.
3. It can be noticed that free carriers which also contribute to PB initially are more efficient on inducing transient absorption than excitons. [23] Then, the free carriers are converted into excitons accompanied by the signal's fast decay, during which the initial free carriers could produce large expansion transport, while excitons will not. These free-carrier spatiotemporal dynamics lead to the ultrafast expansion of PB signal, but then decline the PB generated by those diffused carriers as well. Due to the retention of substantial excitons at the center, the proportion between central and surrounding PB will increase, causing a concentration towards the center.
The low-energy PA feature peaks at 1.92 eV, ~60 meV lower than the exciton resonance.
This feature exhibits roughly the same kinetics as that of the PB feature ( Figure S4), indicating that it should be originated from the bound exciton. Generally, this low-energy feature may be caused by two effects: biexciton absorption [24] or trapped-state exciton formation. Here we believe that biexciton formation is the dominant mechanism due to the diffusion dynamics shown in Figure S8b. Specifically, since the exciton density at the pump center is much higher than that at surroundings, biexciton generation efficiency is much higher at the pump center, leading to an initial contraction of the PA feature. In contrast, the center's high exciton density will saturate the local defect state, resulting in a smaller probe absorption compared to that at surroundings, and thus an initial expansion rather than contraction occurs for the PA feature. According to the experimental results in Figure S8b, an initial contraction with =-240 cm 2 s -1 is found, indicating that the PA feature at 1.92 eV is caused by the generation of biexcitons. The 60 meV energy offset is roughly the same as the biexciton binding energy, [25] further supporting this ascription.  The pump and probe wavelengths are 400 and 620 nm, respectively. These two values are obtained by testing a defective monolayer CVD-growth WS 2 sample in which the carrier transport is limited and the temporal dynamics are faster than multilayers. Even so, here the actual resolution of the system will be better than the measured value which is a convolution between that and the carrier dynamic.   and slow ( slow =120 ps, slow =49%) decays.